Unary multiple equality sets: The languages of rational matrices

Abstract

Unary multiple equality sets are equality sets of finite sets of homomorphisms, which map into the free monoids over single letter alphabets. These languages can equally well be defined by rational matrices. Unary multiple equality sets have extraordinary language theoretic properties. They coincide with the commutative multiple equality sets, each unary multiple equality set is precisely classified by the rank of its defining matrices, and all fundamental decision questions are polynomially decidable. The least trio containing all unary multiple equality sets equals the class of languages accepted by blind multicounter machines and by simple multihead finite automata. Finally crossing-free unary multiple equality sets are introduced, which vary from unary multiple equality sets in a similar way as the one-sided Dyck set varies from the two-sided Dyck set.